A Rapid Graphical Method for Predicting Infrared Absorption Band Contours W. R. Bradford

E.M.I. Electronics Ltd., Feltham, Middlesex, England. Received 25 May 1965.

The well-known Elsasser model of an infrared absorption band postulates a uniform array of Lorentzian absorption lines equal in both intensity and spacing.ι For such a band model the mean spectral transmittance (smoothing out the individual line structure) may be expressed in terms of two dimensionless parameters, β = 2πα/d and χ = Su/2πα, where α is the line half-width at half-intensity, d the line spacing, S the line in­tensity, and u the quantity of absorber. One form of the transmission function for an Elsasser band is

Fig. 1. "A" chart for the Elsasser band model.

where τ is the mean spectral transmittance and σ a variable of integration. Although this transmission function has been tabulated,2,3 it is generally used in the form of approximations

Fig. 2. "B" chart for the 4.3-μ carbon dioxide band.

September 1965 / Vol. 4, No. 9 / APPLIED OPTICS 1207

Fig. 3. "A" and "B" charts superimposed to give mean spectral transmittance for a path length of 1 cm in CO2 at a pressure of

0.5 atm.

valid for the limiting conditions of strong, weak, and nonover-lapping lines. Plass4 finds that one or other of these approxima­tions is valid for representing absorption with an accuracy of 10% over the entire range of values of β and χ. However, for predicting the amount of energy transmitted through a large quantity of absorber, the approximations are not always suitable; the corresponding uncertainty in transmittance may be much greater than 10% when β and χ are both greater than unity.

A new graphical method of predicting transmittance is de­scribed, which does not depend upon the validity of the approxi­mations. The method is illustrated for the 4.3-μ carbon dioxide band and the Elsasser band model transmission function (1), although it has also been used for other absorption bands and for the statistical band model. The essence of the method is the superposition of two charts (shown in Figs. 1 and 2), their rel­ative position being determined by the values of the path length and pressure of the absorber for which the transmittance is

required. The lower, or "A" chart (Fig. 1) carries curves of constant τ calculated from the transmission function (1) plotted in logarithmic coordinates of β and \/χ. The axes of the "A" chart, however, are labeled pressure, p, instead of β, and path length, l, instead of χ. Appropriate corrections for the changed coordinates are introduced by means of a second transparent chart "B" (Fig. 2), which carries wavenumber-dependent log­arithmic corrections for the required absorption band. These corrections are defined by the relationships

where α0 is the line half-width at a pressure of 1 atm and at the required temperature.

In use, the "B" chart is set with its origin at a point on the "A" chart defined by the required values of p and l (Fig. 3), and transmittance is read directly for each marked wavenumber point on the "B" chart. Visual interpolation between adjacent transmittance curves may be necessary. In effect, the two spectral coefficients in (2) have been introduced by a slide rule action along each coordinate axis. It is possible to note very quickly the minimum transmittance in a band, or the wave-numbers between which the transmittance is less than some specified value. The separate effects of changes in pressure and path length are at once apparent by moving the "B" chart horizontally or vertically.

Some further points should be noted. The "A" chart depends only on the band model and contains no spectral information. The wavenumber range covered on a "B" chart is limited by the complexity of the resulting curve. Generally, a single chart is sufficient for each absorption band. For a given absorption band, the " B " chart can be modified to represent the effect of a change of temperature or spectral resolution. Although both charts have a theoretical foundation, it is possible to construct the "B" chart directly from experimental measurements by an inverse application of the method described. At least two meas­urements, corresponding to both strong and weak absorption are necessary at each wavenumber to give well-defined points. The "B" chart shown in Fig. 2 was constructed in this way from laboratory measurements of Burch et al.5

Acknowledgment is due to J. E. A. Selby for assistance in preparing the charts.

References 1. W. M. Elsasser, Phys. Rev. 54, 126 (1938). 2. L. D. Kaplan, J. Meteorol. 10, 100 (1953). 3. D. Q. Wark and M. Wolk, Monthly Weather Rev. 58, 249

(1960). 4. G. N. Plass, J. Opt. Soc. Am. 50, 868 (1960). 5. D. E. Burch, D. Gryvnak, E. B. Singleton, W. L. France, and

D. Williams, AFCRL-62-698, Air Force Cambridge Research Laboratories, Bedford, Mass. (July 1962).

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